Fourier transform approach to nonperiodic boundary value problems in porous conductive media

Abstract

In this article, we develop an extension of the Fourier transform solution method in order to solve conduction equation with nonperiodic boundary conditions (BC). The periodic Lippmann-Schwinger equation for porous materials is extended to the case of non-periodicity with relevant source terms on the boundary. The method is formulated in Fourier space based on the temperature as unknown, using the exact periodic Green function and form factors to describe the boundaries. Different types of BC: flux, temperature, mixed and combined with periodicity can be treated by the method. Numerical simulations show that the method does not encounter convergence issues due to the infinite contrast and yields accurate results for both local fields and effective conductivity.

Fourier Transform approach to numerical homogenization of periodic media containing sharp insulating and superconductive cracks

“…Based on previous works for porous material (To and Bonnet, 2020;To et al, 2021), the new LS governing equations can be derived when the pore tends to the crack limit. In this case, the temperature jump and the heat flux jump are unknown and can be solved in the new formulation.…”

“…In the present paper, a FFT method dealing with cracks of zero thickness is presented. We note that the case of finite insulating pore can be treated by LS equations derived for skeleton temperature (To and Bonnet, 2020;To et al, 2021). It suggested that the method could be extended to insulating and superconductive cracks.…”

Fourier transform approach to homogenization of frequency-dependent heat transfer in porous media

“…As a result, the integral equations to solve the heat transfer problem with source term can be obtained (To et al , 2021). Depending on the choice of unknowns, we have, say, for gradient e : for the flux j : and for the polarization τ : …”

“…The method can be extended to nonperiodic problems where the source term can be used to impose heat flux and temperature, for example the immersed interface technique (Peskin, 1972;LeVeque and Li, 1994;Wiegmann, 1990). The details of FT implementation using form HFF 33,6 factor and continuous Green tensors can be found in a recent work (To et al, 2021). Another possibility is to use polarization (or eigenstress/strain for elasticity) as done by Gelebart (2020) or Chen et al (2019).…”

“…It is shown that most of iterative schemes fail to satisfy the divergence free and rotational free criteria (Moulinec and Silva, 2014). To overcome those limitations, To and Bonnet (2020) and To et al (2021) recently proposed a class of iteration scheme based on boundary integral equation. Another interesting technique is to use discrete Green tensors and modified wavevectors as derived from finite difference approximation (Willot et al , 2014).…”